On the electrodynamics of moving bodies at low velocities Marc de Montigny, Germain Rousseaux To cite this version: Marc de Montigny, Germain Rousseaux. On the electrodynamics of moving bodies at low velocities. European Journal of Physics, European Physical Society, 2006, 27 (4), p. 755-768. 10.1088/0143- 0807/27/4/007. hal-00016214 HAL Id: hal-00016214 https://hal.archives-ouvertes.fr/hal-00016214 Submitted on 21 Dec 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the electrodynamics of moving bodies at low velocities 1 M. de Montignya,b and G. Rousseauxc aCampus Saint-Jean, University of Alberta 8406 - 91 Street Edmonton, Alberta, Canada T6C 4G9 bTheoretical Physics Institute, University of Alberta Edmonton, Alberta, Canada T6G 2J1 cUniversit´ede Nice Sophia-Antipolis, Institut Non-Lin´eaire de Nice, INLN-UMR 6618 CNRS-UNICE, 1361 route des Lucioles, 06560 Valbonne, France Abstract We discuss the seminal article in which Le Bellac and L´evy-Leblond have identified two Galilean limits of electromagnetism [1], and its modern implica- tions. We use their results to point out some confusion in the literature and in the teaching of special relativity and electromagnetism. For instance, it is not widely recognized that there exist two well defined non-relativistic limits, so that researchers and teachers are likely to utilize an incoherent mixture of both. Recent works have shed a new light on the choice of gauge conditions in classical electromagnetism. We retrieve Le Bellac-L´evy-Leblond’s results by examining orders of magnitudes, and then with a Lorentz-like manifestly co- variant approach to Galilean covariance based on a 5-dimensional Minkowski manifold. We emphasize the Riemann-Lorenz approach based on the vector and scalar potentials as opposed to the Heaviside-Hertz formulation in terms of elec- tromagnetic fields. We discuss various applications and experiments, such as in magnetohydrodynamics and electrohydrodynamics, quantum mechanics, su- perconductivity, continuous media, etc. Much of the current technology where waves are not taken into account, is actually based on Galilean electromag- netism. Key words: Galilean covariance, special relativity, electromagnetism, four-potential. ccsd-00016214, version 1 - 21 Dec 2005 1E-mail: [email protected], [email protected] 1 Introduction The purpose of this article is to emphasize the relevance of Galilean covariance in physics, even nowadays, about one hundred years after Lorentz, Poincar´eand Ein- stein, then facing the apparent incompatibility between Galilean mechanics and the full set of Maxwell equations, have developed a theory that turned into special rel- ativity [2]. Seventy years later, Le Bellac and L´evy-Leblond (LBLL) observed that there exist not only one, but two well-defined Galilean (that is, non-relativistic) limits of electromagnetism: the so-called ‘magnetic’ and ‘electric’ limits [1]. Although spe- cial relativity has superseded Galilean relativity when it comes to the description of high energy phenomena, there exists a wealth of low-energy systems, particularly in condensed matter physics and low-energy nuclear physics, where Galilean covariance should not be ignored. We wish to point out hereafter some confusion which results from not recognizing appropriately the two Galilean limits of electromagnetism. This follows from inacu- rate definitions of non-relativistic covariance, which is why we emphasize at once that the definition of Galilean covariance employed henceforth in this paper rests on its compatibility with the Galilean transformations of space-time (Eq. (6), below). Ex- amples of misleading, though well known, such text presentations are mentioned in [1], and there were many more since then. The fact that one should be careful when dealing with electrodynamics at low velocities has been illustrated, for instance, in Ref. [3]. Let us illustrate this point with a simple example. Under a Lorentz transfor- mation with relative velocity v, the electric and magnetic fields, in vacuum, become ′ v(v·E) E = γ(E + v B)+(1 γ) v2 , ′ 1 × − v(v·B) (1) B = γ(B 2 v E)+(1 γ) 2 , − c × − v respectively. The fact that Galilean covariance is a much more subtle concept than simply taking the v <<c, or γ 1, limit is illustrated by the fact that Eq. (1) then becomes: ≃ E′ = E + v B, ′ 1 × (2) B = B 2 v E, − c × which not only is not compatible with Galilean relativity but, worse, does not even satisfy the composition properties of transformation groups [1, 3]. That is to say, a sequence of such transformations does not have the same form as above. We have organized this article as follows. In Section 2, we recall the main results of LBLL [1] for later reference. In Section 3, we obtain these results using two ar- guments: one based on orders of magnitudes and a recent covariant approach with which the Galilean space-time is embedded into a five-dimensional space. Throughout the paper, we favour the Riemann-Lorenz formulation of electrodynamics, based on the scalar and vector potentials, over the Heaviside-Hertz approach which involved electromagnetic fields [4]. Discussion and applications are in Section 4. Therein we present magnetohydrodynamics and electrohydrodynamics as two not-so-distant 1 cousins which can be associated respectively with the ‘magnetic’ and ‘electric’ Galilean limits of electrodynamics. We show that engineers are used to employ these Galilean limits which they denote as the electro- and magnetoquasistatics. We revisited Feyn- man’s proof of the magnetic limit and illustrate the latter within the realm of super- conductivity. Finally, we reassess our current understanding of the electrodynamics of moving bodies by examining the Trouton-Noble experiment in a Galilean context... 2 Galilean electromagnetism The purpose of LBLL was to write down the laws of electromagnetism in a form compatible with Galilean covariance rather than Lorentz covariance. As LBLL put it, such laws could have been devised by a physicist in the mid-nineteenth century [1]. Here, let us retrieve these laws from relativistic kinematics. The Lorentz trans- formation of a four-vector (u0, u), where the four components have the same units, is given by (see chapter 7 of Ref. [5]): ′0 0 1 u = γ u c v u , ′ v− 0 · v (3) u = u γ u +(γ 1) 2 v u, − c − v · where γ 1 , with a relative velocity v. The speed of light in the vacuum ≡ √1−v2/c2 is denoted c. LBLL were the first to observe that this transformation admits two well-defined Galilean limits [1]. One limit is for timelike vectors: u′0 = u0, ′ (4) u = u 1 vu0, − c which, as we shall see, may be related to the so-called electric limit. The second limit is for spacelike vectors: ′0 0 1 u = u c v u, ′ (5) u = u, − · and will be associated to the magnetic limit. As it is well known, the space-time coordinates can be described by timelike vectors only. Indeed, Eq. (4) has the form of Galilean inertial space-time transformations: x′ = x vt, ′ − (6) t = t. Nevertheless, other vectors, such as the four-potential and four-current, may trans- form as one or the other of the two limits. An example of the subtlety of non-relativistic kinematical covariance is that it is quite common to neglect to enforce the condition that a non-relativistic limit involves not only low-velocity phenomena, but also large timelike intervals then one obtains different kinematics, referred to as Carroll kinematics [6]. In other terms, a Galilean 2 world is one within which units of time are naturally much larger than units of space. The existence of events physically connected by large spacelike intervals would imply loss of causality, among other things. Other such kinematics, each one being some limit of the de Sitter kinematics, have been classified in [7]. The situation is similar with electric and magnetic fields. One needs to compare the module of the electric field E to c times the module of the magnetic field, i.e. cB. When the magnetic field is dominant, Eq. (1) reduces to a transformation referred to as the magnetic limit of electromagnetism: ′ Em = Em + v Bm, Em << cBm, ′ × (7) Bm = Bm. The other alternative, where the electric field is dominant, leads to the electric limit: ′ Ee = Ee, Ee >> cBe, ′ 1 (8) B = B 2 v E . e e − c × e Indeed, the approximations Ee/c >> Be and v << c together imply that Ee/v >> Ee/c >> Be so that we take Ee >> vBe in Eq. (1). Such an analysis of orders of magnitude is described in the next section. From the Galilean transformations of space-time, Eq. (6), we find ′ = , ∂ ′ = ∂ + v . (9) ∇ ∇ t t · ∇ The fields transformations in the magnetic limit of Eq. (7) are clearly compatible with the use of Eq. (9) together with the transformations of the four-potential (V, A): ′ Vm = Vm v Am, ′ − · (10) Am = Am, (note the similarity with Eq. (5)) where E = V ∂ A , B = A . (11) m −∇ m − t m m ∇ × m Similarly, the electric limit of Eq. (8) may be obtained from Eq. (9) and the trans- formations of the four-potential: ′ Ve = Ve, ′ v (12) A = A 2 V . e e − c e This equation is similar to Eq. (4). Now, however, the fields are related to the four-potential by E = V , B = A .